Centralizer - Imported Knowlpedia Content

Let $G$ be a group and let $S\subseteq G$ be a subset. The centralizer of $S$ in $G$ is

C_G(S) \;=\; \{\,g\in G : gs=sg \text{ for all } s\in S\,\}.

This is a subgroup of $G$ . For a single element $x\in G$ , one writes $C_G(x)$ for $C_G(\{x\})$ .

Remarks

Centralizers organize commutation in a group and control conjugacy classes (e.g. via orbit–stabilizer for the conjugation action). The center satisfies $Z(G)=C_G(G)$ .

Examples

If $G$ is abelian, then $C_G(S)=G$ for every subset $S$ .
In $S_3$ , $C_{S_3}((12))=\{e,(12)\}$ .
In $S_3$ , $C_{S_3}((123))=\{e,(123),(132)\}$ .